cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A201908 Irregular triangle of 2^k mod (2n-1).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 11, 1, 2, 4, 8, 16, 9
Offset: 1

Views

Author

T. D. Noe, Dec 07 2011

Keywords

Comments

The length of the rows is given by A002326. For n > 1, the first term of row n is 1 and the last term is n. Many sequences are in this one: starting at A036117 (mod 11) and A070335 (mod 23).
Row n, for n >= 2, divided elementwise by (2*n-1) gives the cycles of iterations of the doubling function D(x) = 2*x or 2*x-1 if 0 <= x < 1/2 or , 1/2 <= x < 1, respectively, with seed 1/(2*n-1). See the Devaney reference, pp. 25-26. D^[k](x) = frac(2^k/(2*n-1)), for k = 0, 1, ..., A002326(n-1) - 1. E.g., n = 3: 1/5, 2/5, 4/5, 3/5. - Gary W. Adamson and Wolfdieter Lang, Jul 29 2020.

Examples

			The irregular triangle T(n, k) begins:
n\k  0 1 2 3  4  5  6  7 8  9 10 11 12 13 14 15 16 17 ...
---------------------------------------------------------
1:   0
2:   1 2
3:   1 2 4 3
4:   1 2 4
5:   1 2 4 8  7  5
6:   1 2 4 8  5 10  9  7 3  6
7:   1 2 4 8  3  6 12 11 9  5 10  7
8:   1 2 4 8
9:   1 2 4 8 16 15 13  9
10:  1 2 4 8 16 13  7 14 9 18 17 15 11  3  6 12  5 10
... reformatted by _Wolfdieter Lang_, Jul 29 2020.

References

  • Robert L. Devaney, A First Course in Chaotic Dynamical Systems, Addison-Wesley., 1992. pp. 24-25

Links

Crossrefs

Cf. A002326, A201909 (3^k), A201910 (5^k), A201911 (7^k).
Cf. A000034 (3), A070402 (5), A069705 (7), A036117 (11), A036118 (13), A062116 (17), A036120 (19), A070347 (21), A070335 (23), A070336 (25), A070337 (27), A036122 (29), A070338 (33), A070339 (35), A036124 (37), A070340 (39), A070348 (41), A070349 (43), A070350 (45), A070351 (47), A036128 (53), A036129 (59), A036130 (61), A036131 (67), A036135 (83), A036138 (101), A036140 (107), A201920 (125), A036144 (131), A036146 (139), A036147 (149), A036150 (163), A036152 (173), A036153 (179), A036154 (181), A036157 (197), A036159 (211), A036161 (227).

Programs

  • GAP
    R:=List([0..72],n->OrderMod(2,2*n+1));;
Flat(Concatenation([0],List([2..11],n->List([0..R[n]-1],k->PowerMod(2,k,2*n-1))))); # Muniru A Asiru, Feb 02 2019
  • Mathematica
    nn = 30; p = 2; t = p^Range[0, nn]; Flatten[Table[If[IntegerQ[Log[p, n]], {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, 1, nn, 2}]]

Formula

T(n, k) = 2^k mod (2*n-1), n >= 1, k = 0, 1, ..., A002326(n-1) - 1.
T(n, k) = (2*n-1)*frac(2^k/(2*n-1)), n >= 1, k = 0, 1, ..., A002326(n-1) - 1, with the fractional part frac(x) = x - floor(x). - Wolfdieter Lang, Jul 29 2020

A201912 Irregular triangle of 2^k mod prime(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1, 2, 4, 8, 16, 3, 6, 12, 24
Offset: 1

Views

Author

T. D. Noe, Dec 17 2011

Keywords

Comments

The row lengths are in A014664. For n > 1, the first term of each row is 1 and the last term is 2*prime(n)-1, which is A006254. Many sequences are in this one.

Examples

			The first 11 rows are:
2:  0;
3:  1, 2;
5:  1, 2, 4, 3;
7:  1, 2, 4;
11: 1, 2, 4, 8,  5, 10,  9,  7,  3,  6;
13: 1, 2, 4, 8,  3,  6, 12, 11,  9,  5, 10,  7;
17: 1, 2, 4, 8, 16, 15, 13,  9;
19: 1, 2, 4, 8, 16, 13,  7, 14,  9, 18, 17, 15, 11,  3,  6, 12,  5, 10;
23: 1, 2, 4, 8, 16,  9, 18, 13,  3,  6, 12;
29: 1, 2, 4, 8, 16,  3,  6, 12, 24, 19,  9, 18,  7, 14, 28, 27, 25, 21, 13, 26, 23, 17, 5, 10, 20, 11, 22, 15;
31: 1, 2, 4, 8, 16;

Links

Crossrefs

Cf. A062117, A201908, A201909, A201910, A201911.
Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), A269266 (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67), A036135 (p=83), A036138 (p=101), A036140 (p=107), A036144 (p=131), A036146 (p=139), A036147 (p=149), A036150 (p=163), A036152 (p=173), A036153 (p=179), A036154 (p=181), A036157 (p=197), A036159 (p=211), A036161 (p=227).

Programs

  • GAP
    P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(2,P[n]));;
Flat(Concatenation([0],List([2..10],n->List([0..R[n]-1],k->PowerMod(2,k,P[n]))))); # Muniru A Asiru, Feb 01 2019
  • Mathematica
    nn = 10; p = 2; t = p^Range[0,Prime[nn]]; Flatten[Table[If[Mod[n, p] == 0, {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, Prime[Range[nn]]}]]

A187532 a(n) = 4^n mod 19.

Original entry on oeis.org

1, 4, 16, 7, 9, 17, 11, 6, 5, 1, 4, 16, 7, 9, 17, 11, 6, 5, 1, 4, 16, 7, 9, 17, 11, 6, 5, 1, 4, 16, 7, 9, 17, 11, 6, 5, 1, 4, 16, 7, 9, 17, 11, 6, 5, 1, 4, 16, 7, 9, 17, 11, 6, 5, 1, 4, 16, 7, 9, 17, 11, 6, 5, 1, 4, 16, 7, 9, 17, 11, 6, 5, 1, 4, 16, 7, 9, 17, 11, 6, 5
Offset: 0

Views

Author

M. F. Hasler, Mar 10 2011

Keywords

Comments

Period 9: repeat (1,4,16,7,9,17,11,6,5).
Also continued fraction expansion of (13140908+sqrt(323139488118562))/24969762. - Bruno Berselli, Sep 09 2011

Links

Crossrefs

Cf. A036120, A070342, A070373, A070395, A070421.

Programs

  • Magma
    [4^n mod 19 : n in [0..80]]; // Vincenzo Librandi, Sep 09 2011
  • Mathematica
    PowerMod[4, Range[0, 100], 19]  (* or *)
PadRight[{}, 100, {1, 4, 16, 7, 9, 17, 11, 6, 5}] (* Paolo Xausa, Mar 17 2024 *)
  • PARI
    a(n)=lift(Mod(4,19)^n) \\ Charles R Greathouse IV, Mar 22 2016

Formula

a(n+9) = a(n).
G.f.: (1 + 4*x + 16*x^2 + 7*x^3 + 9*x^4 + 17*x^5 + 11*x^6 + 6*x^7 + 5*x^8)/((1-x)*(1+x+x^2)*(1+x^3+x^6)). - Bruno Berselli, Sep 09 2011

A269266 a(n) = 2^n mod 31.

Original entry on oeis.org

1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1
Offset: 0

Views

Author

Vincenzo Librandi, Mar 31 2016

Keywords

References

  • Continued fraction expansion of (1651+sqrt(3236405))/2386. - Bruno Berselli, Mar 31 2016

Links

Crossrefs

Cf. A201912 (11th row of the triangle).
Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), this sequence (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67).

Programs

  • GAP
    List([0..70],n->PowerMod(2,n,31)); # Muniru A Asiru, Jan 30 2019
  • Magma
    [Modexp(2, n, 31): n in [0..100]];
  • Magma
    &cat [[1,2,4,8,16]^^20] // Bruno Berselli, Mar 31 2016
  • Mathematica
    PowerMod[2, Range[0, 100], 31]
  • PARI
    a(n)=2^(n%5) \\ Charles R Greathouse IV, Mar 31 2016
  • PARI
    x='x+O('x^99); Vec((1+2*x+4*x^2+8*x^3+16*x^4)/(1-x^5)) \\ Altug Alkan, Mar 31 2016
  • Python
    for n in range(0,100):print(2**n%31) # Soumil Mandal, Apr 03 2016
  • Python
    def A269266(n): return pow(2,n,31) # Chai Wah Wu, Jan 03 2022
  • Sage
    [2^mod(n,5) for n in (0..100)] # Bruno Berselli, Mar 31 2016

Formula

G.f.: (1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4)/(1 - x^5).
a(n) = a(n-5).
a(n) = 2^(n mod 5). - Bruno Berselli, Mar 31 2016

A008832 Discrete logarithm of n to the base 2 modulo 19.

Original entry on oeis.org

0, 1, 13, 2, 16, 14, 6, 3, 8, 17, 12, 15, 5, 7, 11, 4, 10, 9
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Equivalently, a(n) is the multiplicative order of n with respect to base 2 (modulo 19), i.e., a(n) is the base-2 logarithm of the smallest k such that 2^k mod 19 = n. - Jon E. Schoenfield, Aug 13 2021

Examples

			From _Jon E. Schoenfield_, Aug 13 2021: (Start)
Sequence is a permutation of the 18 integers 0..17:
   k     2^k  2^k mod 19
  --  ------  ----------
   0       1           1  so a(1)  =  0
   1       2           2  so a(2)  =  1
   2       4           4  so a(4)  =  2
   3       8           8  so a(8)  =  3
   4      16          16  so a(16) =  4
   5      32          13  so a(13) =  5
   6      64           7  so a(7)  =  6
   7     128          14  so a(14) =  7
   8     256           9  so a(9)  =  8
   9     512          18  so a(18) =  9
  10    1024          17  so a(17) = 10
  11    2048          15  so a(15) = 11
  12    4096          11  so a(11) = 12
  13    8192           3  so a(3)  = 13
  14   16384           6  so a(6)  = 14
  15   32768          12  so a(12) = 15
  16   65536           5  so a(5)  = 16
  17  131072          10  so a(10) = 17
  18  262144           1
but a(1) = 0, so the sequence is finite with 18 terms.
(End)

References

  • Carl Friedrich Gauss, "Disquisitiones Arithmeticae", Yale University Press, 1965; see p. 37.
  • I. M. Vinogradov, Elements of Number Theory, p. 221.

Links

Crossrefs

Cf. A036120.

Programs

  • Maple
    [ seq(mlog(n,2,19), n=1..18) ];
  • Mathematica
    a[1]=0; a[n_]:=MultiplicativeOrder[2, 19, {n}]; Array[a, 18] (* Vincenzo Librandi, Mar 21 2020 *)
  • PARI
    a(n) = znlog(n, Mod(2, 19)); \\ Kevin Ryde, Aug 13 2021
  • Python
    from sympy.ntheory import discrete_log
def a(n): return discrete_log(19, n, 2)
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Aug 13 2021

Formula

2^a(n) == n (mod 19). - Michael S. Branicky, Aug 13 2021

Extensions

Offset corrected by Jon E. Schoenfield, Aug 12 2021
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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